M06-16

Official Solution:


If \(\frac{a}{b} = \frac{x}{y}\) and \(\frac{a}{y} = \frac{b}{x}\), where \(a\), \(b\), \(x\), and \(y\) are non-zero integers, which of the following must be true?

I. \(\frac{x}{y}=-1\)

II. \(x=y\)

III. \(|x| = |y|\)


A. I only
B. II only
C. III only
D. I and III only
E. I, II and III


We start by expressing the second equation as \(\frac{a}{b} = \frac{y}{x}\), so that \(x\) and \(y\) are on the same side. Hence, we have \(\frac{a}{b} = \frac{x}{y}\) and \(\frac{a}{b} = \frac{y}{x}\). Therefore, \(\frac{x}{y}= \frac{y}{x}\), which simplifies to \(x^2=y^2\). Taking the square root of both sides gives \(|x| = |y|\). Hence, only option III is always true.

To see why options I and II are not always true, consider that if \(x\) and \(y\) have the same sign, then from \(|x| = |y|\) we get \(x=y\), and in this case, option I is not true. For example, consider \(a=b=x=y=1\). Conversely, if \(x\) and \(y\) have opposite signs, then from \(|x| = |y|\) we get \(x=-y\), and in this case, option II is not true. For example, consider \(a=x=1\) and \(b=y=-1\).


Answer: C